6.2 Graphs of the Other Trigonometric Functions

Graph of tangent function

The tangent function $y = \tan x$ behaves very differently from sine and cosine. Instead of oscillating between fixed values, it shoots from $-\infty$ to $+\infty$ within each period, creating a graph with vertical asymptotes and a distinctive S-shaped curve.

Here are the core properties:

• Period: $\pi$ (the pattern repeats every $\pi$ units along the x-axis)
• Domain: All real numbers except odd multiples of $\frac{\pi}{2}$, where vertical asymptotes occur
• Range: All real numbers ($-\infty$ to $+\infty$)
• Zeros: The graph crosses the x-axis at integer multiples of $\pi$ (i.e., $x = 0, \pm\pi, \pm2\pi, \ldots$)

Within one period, say from $x = -\frac{\pi}{2}$ to $x = \frac{\pi}{2}$, the function increases from $-\infty$ to $+\infty$. As $x$ approaches $\frac{\pi}{2}$ from the left, the function heads toward $+\infty$. As $x$ approaches $-\frac{\pi}{2}$ from the right, it heads toward $-\infty$. This behavior comes directly from the definition $\tan x = \frac{\sin x}{\cos x}$: the asymptotes happen wherever $\cos x = 0$.

Variations of tangent function

Transformations follow the general form $y = a\tan(b(x - h)) + k$. Each parameter controls a specific change:

• Vertical shift ($k$): $y = \tan x + k$ shifts the graph up by $k$ units (or down if $k < 0$)
• Horizontal shift ($h$): $y = \tan(x - h)$ shifts the graph right by $h$ units (or left if $h < 0$)
• Vertical stretch/compression ($a$): $y = a\tan x$ stretches the graph vertically by a factor of $|a|$ when $|a| > 1$, and compresses it when $0 < |a| < 1$. If $a < 0$, the graph also reflects across the x-axis.
• Horizontal stretch/compression ($b$): $y = \tan(bx)$ changes the period to $\frac{\pi}{|b|}$. When $|b| > 1$, the graph compresses horizontally (shorter period). When $0 < |b| < 1$, it stretches (longer period).

The $b$-value is the one students most often mix up. Remember: a larger $|b|$ means a smaller period, not a larger one.

Secant and cosecant graphs

Secant and cosecant are the reciprocals of cosine and sine, respectively: $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$. Their graphs look like a series of U-shaped curves (opening up and down) separated by vertical asymptotes.

Both share these properties:

• Period: $2\pi$
• Range: $(-\infty, -1] \cup [1, \infty)$. The function values are never between $-1$ and $1$.

Where they differ is in the placement of their asymptotes:

Cotangent function analysis

The cotangent function $y = \cot x$ is defined as $\cot x = \frac{\cos x}{\sin x}$. It's similar to tangent in many ways but with some key differences in its graph.

• Period: $\pi$
• Domain: All real numbers except integer multiples of $\pi$ (where $\sin x = 0$)
• Range: All real numbers
• Zeros: Odd multiples of $\frac{\pi}{2}$ (i.e., $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \ldots$)

The big visual difference from tangent: within each period, cotangent decreases from $+\infty$ to $-\infty$, while tangent increases. On the interval $(0, \pi)$, the graph comes down from $+\infty$ near $x = 0$, passes through zero at $x = \frac{\pi}{2}$, and drops toward $-\infty$ near $x = \pi$.

Transformations of reciprocal functions

The same transformation rules that apply to tangent also apply to secant, cosecant, and cotangent. The general forms are:

• Secant: $y = a\sec(b(x - h)) + k$
• Cosecant: $y = a\csc(b(x - h)) + k$
• Cotangent: $y = a\cot(b(x - h)) + k$

In each case, $a$ controls vertical stretch/reflection, $b$ changes the period, $h$ shifts horizontally, and $k$ shifts vertically. The new period is $\frac{2\pi}{|b|}$ for secant and cosecant, and $\frac{\pi}{|b|}$ for cotangent. The asymptotes shift along with the horizontal transformation, so always recalculate their positions after applying $b$ and $h$.

Properties of trigonometric functions

This table summarizes the key properties of all six trig functions. It's worth memorizing.